Question

A supply company manufactures copy machines. The unit cost C (the cost in dollars to make each copy machine) depends on the number of machinesmade. Ifx machines are made, then the unit cost is given by the function C(x)=0.8x? - 400x + 54,748. How many machines must be made to minimizethe unit cost?Do not round your answer

Answer 1

$\begin{gathered} \text{The cost function is given by:} \\ C(x)=0.8x^2-400x+54,748 \end{gathered}$

In order to minimize the unit cost;

$\frac{dC(x)}{dx}=0$

Differentiating the cost function with respect to x, we have:

$\frac{dC(x)}{dx}=1.6x-400+0$

Thus, to minimize the cost function, we have:

$\begin{gathered} 1.6x-400=0 \\ 1.6x=400 \\ x=\frac{400}{1.6} \\ x=250 \end{gathered}$

Hence, 250 machines have to be produced in order to minimize the unit cost